# Noisy Accelerated Power Method for Eigenproblems with Applications

## TL;DR

This paper presents a scalable, noise-tolerant algorithm for efficiently computing dominant eigenvectors of symmetric matrix pairs, combining approximation theory and convex optimization to outperform traditional methods in speed and accuracy.

## Contribution

The paper introduces a novel noisy accelerated power method that decomposes eigenproblem solutions into approximate subproblems, achieving faster convergence with theoretical guarantees.

## Key findings

- Linear running time in input size
- Significant acceleration in canonical correlation analysis
- Empirical results show improved accuracy and speed

## Abstract

This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with strong theoretical performance guarantees. More precisely, the algorithm retains the simplicity of the well-known power method but enjoys the asymptotic iteration complexity of the powerful Lanczos method. Unlike these classic techniques, our algorithm is designed to decompose the overall problem into a series of subproblems that only need to be solved approximately. The combination of good initializations, fast iterative solvers, and appropriate error control in solving the subproblems lead to a linear running time in the input sizes compared to the superlinear time for the traditional methods. The improved running time immediately offers acceleration for several applications. As an example, we demonstrate how the proposed algorithm can be used to accelerate canonical correlation analysis, which is a fundamental statistical tool for learning of a low-dimensional representation of high-dimensional objects. Numerical experiments on real-world data sets confirm that our approach yields significant improvements over the current state-of-the-art.